In control theory, a system is some mathematical relation between an input and an output. Systems are usually represented as a rectangle connecting some input function to an output function. Figure 1 depicts a SISO (single input, single output) system taking in an input function of time $u(t)$ and returning an output function of time $y(t)$.
Figure 1: A SISO system
Generally, ordinary differential equations (ODEs) are the simplest way to represent a system.
Note: dependent variables for ODEs (usually, this is time) are generally positive real numbers
Laplace Transform
We use the Laplace transform to switch between a function of real numbers (for our purposes, lets assume this is time $t$) to a function of some complex variable $s$ (Figure 2). Note that our $s$ variable turns out to denote frequency since our original function is a function of time.
Figure 2: Laplace transfrom for our input function $u(t)$ and our output function $y(t)$ from Figure 1.
Specifically, the expression for the Laplace transfrom of a single-variable function of time $f(t)$ is \begin{equation} F(s) = \mathcal{L} \left[ f(t) \right] = \int_{0}^{\infty} f(t)e^{-st} dt \end{equation}
The following equations are simplified expressions of the Laplace transform for $f(t)$ and its corresponding first and second order ODEs:
Thus, we have defined our mechanical system as a second order ODE and as a transfer function. Figure 5 depicts the corresponding system diagrams in terms of the ODE and in terms of the transfer function.
Figure 5: Two equivalent system diagrams for our mass-spring system. The top system diagram represents our mass-spring system in terms of a second order ODE. The bottom system diagram represents our mass-spring system in terms of a transfer function.
For any right triangle (i.e., a triangle with a $90^\circ$ angle between two of its segments), we can apply the following equations to determine the length of segments that comprise the triangle or the degree of the acute angles in the triangle.
A circle is defined as the set of points lying at a fixed distance from some center point. If we set the center of a circle with a radius $r$ to be the origin of a 2D Cartesian coordiante system, then we can use the Pythagorean theorem to find any set of points $P:(r_x, r_y)$ that lie along the perimeter of our circle.
If we want to know the $r_x$ and $r_y$ coordinates associated with some $0 \leq \theta \leq 2\pi$ radians, we can use our first two SOHCAHTOA equations:
$r_x= r \cos(\theta)$
$r_y = r \sin(\theta)$
Aside: to convert between radians and degrees, use the equality $\pi \, \mathrm{rad} = 180^\circ$
Also, let’s say there are $n_1$ possibilities for the $r=1^{\mathrm{st}}$ item, $n_2$ possibilities for the $r=2^{\mathrm{nd}}$ item, …, and $n_r$ possibilities for the $r=r^{\mathrm{th}}$ item
Then, the total number of possibilities for all the different $r$ items is $(n_1)(n_2) \cdot … \cdot (n_r)$
Permulations
A permutation is an ordered set of $r$ items selected from a (larger) set of $n$ items without replacement
Note: these are a special case of the multiplication rule
Specifically, we use permutations when the order in which we choose an item from a set is relevant
Theorem: the number of $n$ distinct items selected $r$ at a time without replacement is… \begin{equation} \frac{n!}{(n-r)!} \end{equation}
Combinations
A combination is an unordered set of $r$ items taken from a (larger) set of $n$ objects without replacement
Theorem: the number of combinations of $r$ items taken without replacement from a (larger) set of $n$ distinct objects is… \begin{equation} \binom{n}{r} = \frac{n!}{(n-r)!r!} \end{equation}
Random Variables
Two different kinds of random variables:
Discrete random variables: characterized by a probability mass function
Continuous random variables: characterized by a probability density function
**CORRECTION: As pointed out by adityaguharoy, there acutally random variables/distributions that are both discrete and continuous (e.g., the cumulative distribution function). Hopefully I’ll remember update this post with more details 🙂
From Special Relativity – Part 1, we know that the speed of light is constant in all inertial frames of reference
Michelson-Morely experiment (1887) showed that the wave-nature of light is not necessarily a mechanical wave causing a disturbance in a medium
In fact, the results of this experiment suggested that there is no medium through which light travels as a mechanical wave
However, it would be a few more decades until the wave-particle duality nature of light was explained in terms of electromagnetic waves
Some Newtonian Physics
Frame of reference (FoR) = an abstract coordinate system that is uniquely fixed (i.e., located and oriented) based on a set of physical reference points
Inertial frame of reference = a FoR in which objects that have a net force of zero acting on them are not accelerated (i.e., they are at rest or move at a cohnstant velocity without changing directions)
In other words, any object that “lives” inside an inertial FoR only accelerates when a physical force is applied to it
Also sometimes called a Galiliean reference frame
Importantly, Einstein’s theory of special relativity only applies for inertial FoRs traveling at a constant velocity relative to one another
A few decades later, Einstein developed general relativity, which applies to inertial reference frames that are accelerating relative to one another
Notes:
For simplicity, we will only consider 1D movement (specifically, movement in the horizontal direction)
As we do more advance physics, we shouldn’t necessarily think as time “driving” position (i.e., graphing an objects position as a function of time)
Although it may feel counterintuitive at first (in most contexts, we graph time on the horizontal axis), but in the context of these “path-time” and later spacetime diagrams, we are going to plot position on the horizontal axis and time on the vertical axis
Maybe changes in position are what drive time!
Constructing our Thought Experiment
Consider a scenario where an observer named Donald Trump is sitting in a spaceship that is drifting through space at a constant velocity
Furthermore, let’s define $S:(x,t)$ as an inertial reference frame in 1D space, whose $x=0$ coordinate corresponds with Trump’s position at any given time $t$
Thus, from Trump’s perspective, his position is constant at the $x=0 \mathrm{m}$ in the $S$-frame
Here is a Newtonian path-time diagram for Trump’s position in the $S$-frame (each $T_i$-th point represents Trump’s position in the $S$-frame at $t=i$ seconds)
Figure 1: Newtonian path-time diagram of Trump’s position in $S:(x,t)$
Now, let’s introduce a second observer named Barack Obama who is sitting in a different spaceship that is traveling at half the speed of light in the positive $x$-direction
We can define a new inertial FoR called $S'(x’,t’)$ whose $x’=0$ coordinate corresponds with Obama’s position at any given time $t’$
Note: we can classify this as an inertial FoR because it is moving at a constant velocity $v=0.5c$ (where $c = 3 \cdot 10^8 \mathrm{m/s}$ is the speed of light) relative to another inertial FoR (i.e., the $S$-frame, which is defined by Trump’s position)
Thus, from Obama’s perspective, his position is constant at the $x’=0 \mathrm{m}$ in the $S’$-frame
Here is a Newtonian path-time diagram for Obama (each $O_i$-th point represents Obama’s position in the $S’$-frame at time $t’=i$ seconds)
Figure 2: Newtonian path-time diagram of Obama’s position in $S’:(x’,t’)$
Transforming Between Different Intertial FoRs
Aside: The Galilean Transformation
Newtonian/classical physics uses the Galilean transformation to transform between points of view (i.e., coordinate systems) for observers/objects “observing” the same event $E$ from different inertial FoRs
Put more simply, Newtonian/classical physics assumes the Galilean transformation accurately describes transforming between different coordinate systems for different intertial FoRs
For example, if we wanted to transform between the coordinates describing where/when an event occurs in the $S$-frame to coordinates in the $S’$-frame, we would use the following equations:
$\begin{matrix} t’ = t \\ x’ = x – vt \end{matrix}$ where $v=$ the velocity of the $S’$-frame relative to the $S$-frame
The invariance of spatial distances is the most significant property in Galilean transformations (i.e., $\Delta x = \Delta x’$)
This property eventually breaks down for speeds near the speed of light
Galiliean physics also relies on the concept of absolute time
Absolute time = the time difference between two events $E_1$ and $E_2$ is the same in all inertial FoRs
Take-home-messages about the Galielean transformation:
Newtonian/classical physcis assumes space and time are absolute in all FoRs
However, for this to hold true, the speed of light should vary among inertial FoRs moving at different velocities relative to one another
As it turns out, the speed of light is always observed to be $c \approx 3\cdot 10^8 \mathrm{m/s}$
Consequently, Newtonian/classical physics “breaks down” for different inertial FoRs traveling at velocities $v \approx c$ relative to one another
Specifically, a unit length for both the time and space axes in one inertial FoR (e.g., $S:(x,t)$ or Trump’s experience of the passage of time and the distance of an event relative to himself) will differ from the unit lengths of these axes in another inertial FoR traveling at a different velocity relative to the first FoR (e.g., $S’:(x’,t’)$ or Obamas experience of the passage of time and the distance of an event relative to himself)
Also, I should probably note that the primary goal of this post is to show that the Galilean transformation does not hold for inertial FoRs traveling at or near the speed of light
Overlaying Path-Time Diagrams Using the Galilean Transformation
Assuming the Galilean transformation holds, let’s overlay the path-time diagram for Obama in the $S’$-frame onto Trump’s path-time diagram in the $S$-frame where Obama’s spaceship passes Trump’s spaceship at $t=0 \mathrm{s}$:
Figure 3: The $S’$-frame overlayed on top of the $S$-frame
This path-time diagram shows Obama’s position changing relative to Trump’s, where Trump’s position is assumed to be constant
Since we are assuming the Galiliean transformation accurately transforms the $S$ and $S’$ coordinate systems, we drew the $t’$ axis so that $t’=t \, \, \forall t’$ and $x’=x$ at $t=0$
Similarly, Trumps path-time diagram overlayed on top of Obama’s would look something like this:
Figure 4: The $S$-frame overlayed on top of the $S’$-frame
Let’s add some more obersvers/events to our thought experiment
Suppose there are two other spaceships, one containing George Bush and the other containing Bill Clinton
Assume the following $S$-frame coordinates for the spaceships
Let’s also assume the velocity of these spaceships are the same as Obama’s
Thus, from Trump’s perspective (i.e., the $S$-frame), these spaceships are traveling at half the speed of light in the positive $x$-direction
From Obama’s perspective (i.e., the $S’$-frame), these spaceships remain at a fixed distance from his spaceship
Here is a path-time diagram of the $S’$-frame overlayed on top of the $S$-frame for these events:
Figure 5: The same path-time diagram depicted in Figure 3, but including Bush’s and Clinton’s path-time trajectories (green). Here, $B_i$ vertices denote Bush’s spaceship at time $t=t’=i$ seconds and $C_i$ vertices denote Clinton’s spaceship at time $t=t’=i$ seconds
Now, suppose Trump turns on a light located on the outersurface of his spaceship at $t=0$
If we draw a path-time diagram in the $S$-frame for the first photon emitted from this light, it would look something like this:
Figure 6: $S$-frame with a path-time trajectory for Trump and a path-time trajectory for the photon emitted at $t=0$
So far, everything looks alright, but if we try to overlay these sequence of events with the $S’$-frame and path-time trajectories for Bush and Clinton, we will see that things start to get contradictory with observations from nature (e.g., Michelson-Moorley experiment)
Figure 7: The same path-time diagram depicted in Figure 5, but including a path-time trajectory for a photon emitted from the $S$-frame at $t=0 \mathrm{s}$ and $x=0 \mathrm{m}$
Consider the photon emitted from the flashlight Trump turned on at $t=0$ seconds (orange line in Figure 7)
Note: the velocity of the photon (that is, light) is $c=3\cdot 10^8 \mathrm{m/s}$
At $t=2 \mathrm{s}$, the photon will have traveled $6 \cdot 10^8 \mathrm{m}$ relative to Trump (i.e., in the $S$-frame)
Using the Galilean transformation and assuming the velocity of the $S’$-frame relative to the $S$-frame is $v=0.5c$, the position of the photon at $t=t’=2 \mathrm{s}$ is…
Thus, from Obama’s point of view (i.e., the $S’$-frame), the velocity of the photon relative to the $S’$-frame seems to be $1.5 \cdot 10^9 \mathrm{m/s}$ (i.e., half the speed of light) in the positive $x$-direction
However, we also know that in reality, regardless of which inertial frame of reference we are in, the speed of light is always the same (i.e., the speed of light should be $3 \cdot 10^8 \mathrm{m/s}$ in both the $S$ and $S’$-frames
Our classical/Newtonian/Galiliean understanding of the universe starts to break down!
Any time we make a prediction and it is not observed in nature, it means our conception of the universe is not complete (Note: it still isn’t complete today!)
We need a new way to conceptualize/visualize path-time diagrams
More content, editing, and diagrams coming soon! 😀 Also, I apologize if anyone takes offense to my version of Einstien’s thought experiement. I’m just trying to keep things interesting becuase this material can be a real mental “climb” if you know what I mean. Plus, I figure most people know who Trump, Obama, Bush, and Clinton are, so it applies to most people (I was going to use Game of Thrones characters, but not everyone knows who they are)
Personal remarks/thoughts:
Should we take into account the biological processes involved in the processing of time, space, and light sensation/perception?
10/5/18 Update:
Photometry: the science of the measurement of light in terms of percieved brightness with respect to the human eye (wiki – photometry)
Note: this is distinct from radiometry – the science of measuring of radiant energy (which includes the light being measured in photometry)
Thought: Do the limitations/bounds of what an organism can/cant percieve have anything to do with different groups of spectral lines for a given element?
Sources
Reiher, M., & Wolf, A. (2015). Relativistic quantum chemistry: The fundamental theory of molecular science. Weinheim: Wiley-VCH Verlag GmbH & KGaA.
In physics, waves are usually associated with the transmission of enegry between different points in space
Three Types of Waves
Mechanical waves
Electromagnetic waves
Matter waves
Mechanical waves
Mechanical waves are propagations of a disturbance thorugh a material medium due to periodic motion of particles that comprise the medium from their mean positions
Existence of a medium is essential for propagation of a mechanical wave
Propagation occurs due to properties of the medium such as elasticity and inertia
Energy and momentum propagate via motion of particles in the medium BUT the overall medium remains in its original position
Two main types of mechanical waves:
Transverse waves = the vibration of particles in the medium occurs perpendicular to the direction of wave propagation (e.g., vibration on a string)
Longitudinal waves = the vibration of particles in the medium occurs parallel to the direction of wave propagation (e.g., oscillations in a spring, internal water waves, tsunamis, sound waves etc.)
Electromagnetic waves
Electromagnetic waves = periodic distortions in electric and magnetic fields
Two components: an electric component and a magnetic component
Initiated when charged particles (e.g., electrons) begin vibrating due to various forces acting on them
The vibration of these charged particles then results in the emission of energy called electromagnetic radiation
Importantly, electromagnetic waves do NOT require a medium to travel through
As far as we know, this property is unique to electromagnetic waves
Other properties:
Travel at the speed of light ($3 \cdot 10^8 \mathrm{m/s}$) in a vacuum
Can be polarized
Transverse in nature
Propagate out from their source (i.e., the vibrating particles)
Oscillations of waves occur perpendicular to the direction they are propagating/traveling
Furthermore, in the case of electromagnetic waves, oscillations in the magnetic component occur in a direction perpendicular to oscillations in the electric componet
No medium required
All EM waves have momentum (thus, they have kinetic energy)
Matter waves
Matter waves (or, de Broglie waves) = depict the wave-like properties of all matter
Assumes wave-particle duality for all matter
Frequency of these waves depends on their kinetic energy
Momentum is not directly (or, inversely) proportional to position of the wave
Miscellaneously-classed waves
Surface waves (or, Rayleigh waves) = can have mechanical or electromagntic nature
